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Regular and Chaotic Dynamics

, Volume 14, Issue 4–5, pp 431–454 | Cite as

Bi-Hamiltonian structures and singularities of integrable systems

  • A. V. Bolsinov
  • A. A. Oshemkov
Articles

Abstract

A Hamiltonian system on a Poisson manifold M is called integrable if it possesses sufficiently many commuting first integrals f 1, … f s which are functionally independent on M almost everywhere. We study the structure of the singular set K where the differentials df 1, …, df s become linearly dependent and show that in the case of bi-Hamiltonian systems this structure is closely related to the properties of the corresponding pencil of compatible Poisson brackets. The main goal of the paper is to illustrate this relationship and to show that the bi-Hamiltonian approach can be extremely effective in the study of singularities of integrable systems, especially in the case of many degrees of freedom when using other methods leads to serious computational problems. Since in many examples the underlying bi-Hamiltonian structure has a natural algebraic interpretation, the technology developed in this paper allows one to reformulate analytic and topological questions related to the dynamics of a given system into pure algebraic language, which leads to simple and natural answers.

Key words

integrable Hamiltonian systems compatible Poisson structures Lagrangian fibrations bifurcations semisimple Lie algebras 

MSC2000 numbers

37K10 37J35 37J20 17B80 

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Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.School of MathematicsLoughborough UniversityLoughboroughUK
  2. 2.Department of Mathematics and MechanicsM.V. Lomonosov Moscow State UniversityMoscowRussia

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